In a message dated 7/12/2000 11:41:42 PM, Kevin.McCann at jhuapl.edu writes:
>I am doing a Legendre expansion of Sin[Pi x] and have, amongst others,
>the following integral
>
>Integrate[Sin[Pi*x]*LegendreP[21,x],{x,-1,1}]
>
>Mathematica just returns the input with the polynomial expanded out. Now,
>everything about the integral is exact. We have a 21st order polynomial
>times the sine, and these integrals are all exact. Why no answer? This
>same integral does work for the 19th and 20th Legendre functions.
>
>However, complications arise even for
>
>Integrate[Sin[Pi*x]*LegendreP[19,x],{x,-1,1}]
>
>Here I do get an exact answer with Pi's and large numbers, but when I do
>N[%] on it, I get an answer of -0.000299144 which is way too large. If
>instead I do N[%,30] on the exact, I get 10^(-14).
Here is a work-around:
soln21 = Integrate[Sin[Pi*x]*LegendreP[21, x], {x, -a, a},
Assumptions -> a > 0] /. a -> 1 // Simplify
1/Pi^21*2*(13113070457687988603440625 -
2025596249561559215165625*Pi^2 + 83648104232906493905625*
Pi^4 - 1435402904039579475000*Pi^6 +
12196233825172897500*Pi^8 - 55437426478058625*Pi^10 +
137561852302875*Pi^12 - 180705224700*Pi^14 +
113565375*Pi^16 - 26565*Pi^18 + Pi^20)
N[soln21, 50]
8.7552179522441867999995890586883880820029597565286863920136\
338316`50*^-17
Extra precision is required for this and your case of n = 19 since the terms
alternate sign.
Bob Hanlon